How to prove the following identity

1. Nov 15, 2012

amitech

1. The problem statement, all variables and given/known data

e_j=g_(jk)e^k

where e_j is a covariant vector base
e^k is a a contravariant vector base
g_(jk) is the covariant metric

2. Relevant equations

3. The attempt at a solution

2. Nov 15, 2012

HallsofIvy

What is the definition of the "covariant metric". How do you go from a covariant to the corresponding contravariant vector and vice-versa?

3. Nov 15, 2012

amitech

Hi, by saying "covariant metic" I mean writing the metric in its covariant form - basically g(contra)=[g(cov)]^(-1)
you can use the metric to lower/raise an index on a vector component but here e_j and e^k are not components but the actual unit vectors.

4. Nov 15, 2012

amitech

Hi, by saying "covariant metic" I mean writing the metric in its covariant form - basically g(contra)=[g(cov)]^(-1)
you can use the metric to lower/raise an index on a vector component but here e_j and e^k are not components but the actual unit vectors.